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The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli. The Pauli group on n {\displaystyle n} qubits, G n {\displaystyle G_{n}} , is the group generated by the operators described above applied to each of n {\displaystyle n} qubits in the tensor product Hilbert space ( C 2 ) ⊗ n {\displaystyle ...
The Fierz identities are also sometimes called the Fierz–Pauli–Kofink identities, as Pauli and Kofink described a general mechanism for producing such identities. There is a version of the Fierz identities for Dirac spinors and there is another version for Weyl spinors. And there are versions for other dimensions besides 3+1 dimensions.
The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension . [2] [3] The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum. The generalized Gell-Mann matrices are Hermitian and traceless by
Since the eight matrices and the identity are a complete trace-orthogonal set spanning all 3×3 matrices, it is straightforward to find two Fierz completeness relations, (Li & Cheng, 4.134), analogous to that satisfied by the Pauli matrices. Namely, using the dot to sum over the eight matrices and using Greek indices for their row/column ...
Arbitrary Clifford group element can be generated as a circuit with no more than (/ ()) gates. [6] [7] Here, reference [6] reports an 11-stage decomposition -H-C-P-C-P-C-H-P-C-P-C-, where H, C, and P stand for computational stages using Hadamard, CNOT, and Phase gates, respectively, and reference [7] shows that the CNOT stage can be implemented using (/ ()) gates (stages -H- and -P ...
In "customary" applications, A is not arbitrary, but a unitary, rotation matrix, whose form is then substantially constrained. Cuzkatzimhut ( talk ) 15:09, 24 March 2015 (UTC) [ reply ] I was looking at similarity transformations on a system of spin- 1 2 {\displaystyle {\tfrac {1}{2}}} particles, and so any spectrum preserving transformation is ...